Exploring CP Violation through Correlations in $B\to\pi K$, $B_d\to\pi^+\pi^-$, $B_s\to K^+K^-$ Observable Space

We investigate allowed regions in observable space of $B\to\pi K$, $B_d\to \pi^+\pi^-$ and $B_s\to K^+K^-$ decays, characterizing these modes in the SM. After a discussion of a new kind of contour plots for the $B\to\pi K$ system, we focus on the mixing-induced and direct CP asymmetries of the decays $B_d\to\pi^+\pi^-$ and $B_s\to K^+K^-$. Using experimental information on the CP-averaged $B_d\to\pi^\mp K^\pm$ and $B_d\to\pi^+\pi^-$ branching ratios, the relevant hadronic penguin parameters can be constrained,implying certain allowed regions in observable space. In the case of $B_d\to \pi^+\pi^-$, an interesting situation arises now in view of the recent B-factory measurements of CP violation in this channel, allowing us to obtain new constraints on the CKM angle $gamma$ as a function of the $B^0_d$--$\bar{B^0_d}$ mixing phase $\phi_d=2\beta$, which is fixed through $A_{CP}^{mix}(B_d\to J/\psi K_S)$ up to a twofold ambiguity. If we assume that $A_{CP}^{mix}(B_d\to \pi^+\pi^-)$ is positive, as indicated by recent Belle data, and that $\phi_d$ is in agreement with the ``indirect'' fits of the unitarity triangle, also the corresponding values for $\gamma$ around $60^\circ$ can be accommodated. On the other hand, for the second solution of $\phi_d$, we obtain a gap around $\gamma\sim60^\circ$. The allowed region in the space of $A_{CP}^{mix}(B_s\to K^+K^-)$ and $A_{CP}^{dir}(B_s\to K^+K^-)$ is very constrained in the SM, thereby providing a narrow target range for run II of the Tevatron and the experiments of the LHC era.